While learning about SVD decomposition in a basic linear algebra course, here's a fun little property of 2x2 matrices I noticed:
$$\mathrm{Let} \ A=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mathrm{and } \ \mathrm{rank}(A)=1;$$ $$\mathrm{It \ follows \ that} \ \lambda_1 = 0 \ \mathrm{and } \ \lambda_2 = a^2+b^2+c^2+d^2.$$
So, for non-invertible matrix $A$, one of the eigenvalues of $A^TA$ is the sum of the squares of the elements of $A$. It is relatively easy to prove by brute-force calculation. Do similar properties exist if $\mathrm{dim}(A) > 2$?
It might if you continue to restrict attention to rank 1 matrices. But if you just allow for any singular matrix, just consider a diagonal 3x3 matrix with exactly one zero on the diagonal.